Gibbs Measures and Quasi-Periodic Solutions for Nonlinear Hamiltonian Partial Differential Equations

Abstract

The purpose of this exposé is to present a summary of some new developments in the theory of Hamiltonian nonlinear evolution equations, more specifically, nonlinear Schrödinger equations. The themes and methods discussed are closely related to classical mechanics. The first topic is the existence of an invariant measure for the flow. This invariant measure is the (properly normalized) Gibbs measure from statistical mechanics and we establish wellposedness of the equation on its support. Those investigations are closely related to the paper [L-R-S]. Results in this direction are obtained in 1D (in the focusing and defocusing case) and in the 2D defocusing case. The second topic concerns the persistency of time periodic and quasi-periodic solutions for Hamiltonian perturbations of linear and integrable equations. We follow a method, the so-called Liapounov-Schmidt decomposition, originating from the works of [C-Wl, 2], rather than the KAM procedure (cf. [Kuk1]). The main advantage of this technique is the fact that it overcomes certain limitations of the KAM scheme, which is necessary to deal in particular with the problems in space-dimension D ≥ 2. This work is a new approach to KAM problems, also in finite dimensional phase space. Persistency results for PDE’s are obtained in the time periodic case in arbitrary dimension and for quasi-periodic solutions when D ≤ 2. The small divisor problems appearing when inverting the linearized operators are related to the works of Frölich and Spencer on the Anderson model.